I have deferred the topic of joint and conditional pdfs for as long as possible. Since we are now equipped to appreciate at least some of its usefulness, its study is now much more interesting than it would have been earlier. The fundamental ‘F’ and ‘t’ distributions of Statistics are introduced in this chapter. Bayesian Statistics is extended to cover multi-parameter situations, and we see how it may be made effective by Gibbs sampling.

One of the reasons for my deferring the topic of joint pdfs is that it can look a bit complicated at first sight. It isn't really. Amongst the theory as it applies to Probability and Statistics, only the ‘Jacobian’ Theorem 249C has any substance; and you can, if you like, take that for granted. (Gibbs sampling certainly has substance too!) What is a little offputting is the notation. Wherever possible, I simplify the appearance of things by using vector notation.

Note. We defer the study of the most important joint pdf, that of the multivariate normal distribution, until Chapter 8.

We start with the simple situation for joint pmfs, which proves a valuable guide to that for joint pdfs.